Yes: Business and Professions Code section 2230.5 requires that any Medical Board accusation be filed within three (3) years after the Board discovers the act or omission that forms the grounds for discipline against a physician, or within seven (7) years after the alleged act or omission actually occurred (whichever occurs first.
NOTE: there are a couple of exceptions to the general statute of limitations described above. An allegation that a physician (or Physician Assistant) procured his or her license by fraud or misrepresentation has no limitation. These limitations also do not apply where a physician deliberately conceals from the Board his or her unprofessional conduct, incompetence, or negligence. Where a minor is involved as a patient or victim, the statute of limitations will toll until the minor reaches adulthood.
In addition, allegations of sexual misconduct by a physician must be filed within three (3) years of discovery by the Board or within ten (10) years of occurrence, whichever occurs first.
Finally, where physician conduct results in a criminal investigation, the statute of limitations is tolled during the investigation if material evidence needed for the accusation is unavailable during that time.

Licensed insurance agents and brokers must report all criminal convictions (misdemeanors and felonies) within 30 days. See California Insurance Code section 1729.2.

California Government Code section 11504 provides that applicants for licensure have a right to a hearing to determine whether they meet the criteria for fitness and licensure. You may request a hearing and a Statement of Issues will be filed and an administrative hearing scheduled with the Office of Administrative Hearings. These hearings are held in San Francisco, Sacramento, Los Angeles or San Diego. Prior to a hearing, the Licensing Division of the Medical Board may offer you a probationary license, or you may be given the option of withdrawing your application. Note: Probationary licenses issued by the California Medical Board at initial licensure do not result in permanently reported discipline on the board's California website so long as you successfully fulfill the terms of probation.  However, board decisions are reported to the National Practitioner Data Bank (NPDB) and a record of probation may remain on that site.  However, the Medical Board does inform the NPDB when probation is completed. In any of the above mentioned cases, you may wish to consult legal counsel, who can represent you at an administrative hearing on a Statement of Issues as well as communicate with the Licensing Division prior to hearing on your behalf.

Yes. Every disciplinary decision and order of the California Medical Board is reported to the National Practitioner Data Bank, a registry of physician disciplinary actions all states report to under federal law. NOTE: if you are subject to disciplinary action in California or another state, you must still report that action to each state where you hold a medical license. California disciplinary actions against physicians or physician assistants are routinely reported by the Medical Board on its internet website.

Can using the letters M.D. after a persons name, or Dr. as a prefix, subject the user to criminal charges?

Surprisingly, the answer is yes! Persons unfamiliar with California Business and Professions Code Section 2054, especially out-of-state physicians and graduates of foreign medical institutions not licensed in California, may not know that using the letters "M.D." "Dr." or the words doctor or physician in their correspondence, business advertising (even non-medical) or communications in California may result in a Medical Board investigation and even misdemeanor criminal charges.

Section 2054 states:

"Any person who uses in any sign, business card, or letterhead, or, in an advertisement, the words "doctor" or "physician," the letters or prefix "Dr.," the initials "M.D.," or any other terms or letters indicating or implying that he or she is a physician and surgeon, physician, surgeon, or practitioner under the terms of this or any other law, or that he or she is entitled to practice hereunder, or who represents or holds himself or herself out as a physician and surgeon, physician, surgeon, or practitioner under the terms of this or any other law, without having at the time of doing so a valid, unrevoked, and unsuspended certificate as a physician and surgeon under this chapter, is guilty of a misdemeanor."

The Medical Board of California takes this statute and its provisions seriously. Even if you have earned your Medical Doctor degree at a recognized institution and are licensed elsewhere, if you do not hold a currently valid California Physician and Surgeon's Certificate, the use of any of these terms can land the user in legal trouble with serious financial, professional and personal consequences. The fact that an individual residing or doing business in California may hold a medical degree and even an out-of-state or foreign medical license does not exempt them from the prohibitions in Section 2054 of the Code.